Path Planning in Discrete Sampled Space Day 25 Path Planning in Discrete Sampled Space 4/19/2019

Connectivity in Discrete Sampled Space a path on a discrete grid is a sequence of moves between connected cells for a square tiling there are two possible definitions of connectivity 4-connectivity 8-connectivity 4/19/2019

4-Connectivity on a 4-connected tiling the distance between two cells is called the taxicab distance, rectilinear distance, L1 distance, L1 norm, city block distance, or Manhattan distance 1 2 3 4 5 6 7 8 9 4/19/2019

Wave-Front Planner the wave-front planner finds a path between a start and goal point in spaces represented as a grid where free space is labeled with a 0 obstacles are labeled with a 1 the goal is labeled with a 2 the start is known 2 1 * start 4/19/2019

Wave-Front Planner starting at the goal cell L := 2 while start cell is unlabelled for each cell C with label L for each cell Z connected to C with label 0 label Z with L+1 L := L + 1 4/19/2019

3 2 1 * start 4/19/2019

4 3 2 1 * start 4/19/2019

5 4 3 2 1 * start 4/19/2019

14 13 12 11 10 9 8 7 6 5 4 3 2 1 * start 4/19/2019

17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 18 19 1 20 * start 4/19/2019

17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 18 19 1 20 21 22 37 38 23 36 24 35 25 34 33 32 29 28 27 26 30 31 50 49 48 46 45 44 43 42 41 40 39 * 47 start 4/19/2019

Wave-Front Planner to generate a path starting from the start point L := start point label while not at the goal move to any connected cell with label L-1 L := L-1 4/19/2019

17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 18 19 1 20 21 22 37 38 23 36 24 35 25 34 33 32 29 28 27 26 30 31 51 50 49 48 46 45 44 43 42 41 40 39 47 start 4/19/2019

Wave-Front Planner advantage: disadvantages: will find a shortest path (in terms of connectivity) between start and goal if a path exists generalizes to higher dimensions disadvantages: path often runs adjacent to obstacles planner searches the entire space with radius R around the goal (where R is the distance between the start and goal) paths restricted by grid connectivity are longer than necessary 4/19/2019

Wave-Front Planner paths restricted by grid connectivity are longer than necessary Manhattan distance = 9 straight line distance = sqrt(16 + 25) = 6.403… 1 2 3 4 5 6 7 8 9 4/19/2019

Planners Using Graph Algorithms A* is a common algorithm in game AI programming and robotics first described in 1968 http://theory.stanford.edu/~amitp/GameProgramming/ A* is the foundation for Theta* Daniel, Nash, Koenig. Theta*: Any-Angle Planning on Grids, Journal of Artificial Intelligence Research, 39, 2010. path planning on a grid where paths are allowed to pass through cells at any angle (not just using 4- or 8-connectivity) 4/19/2019

Potential Functions in continuous space potential functions can be used for path planning a potential function is a differentiable real-valued function U i.e., U assigns a scalar real value to every point in space potential functions you might know gravitational potential electrostatic potential 4/19/2019

Goal Potential the goal potential should be an attractive potential small near the goal large far from the goal monotonically increasing nice too if it is continuously differentiable 4/19/2019

located at a minimum in U Goal Potential consider the quadratic potential “rolls” towards the goal goal located at a minimum in U 4/19/2019

Goal Potential “rolling towards the goal” can be accomplished using gradient descent gradient descent starting at initial configuration, take a small step in the direction opposite to the gradient F until |F| = 0 4/19/2019

Goal Potential notice that the wave-front planner basically works this way it defines a potential where there is only one minimum the minimum is located at the goal it then uses gradient descent to move towards the goal 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 18 19 1 20 21 22 37 38 23 36 24 35 25 34 33 32 29 28 27 26 30 31 51 50 49 48 46 45 44 43 42 41 40 39 47 start 4/19/2019